Sufficient Conditions for a Digraph to be Supereulerian

A (di)graph is supereulerian if it contains a spanning eulerian sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we analyze a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomassé and the first author: if the arc‐connectivity of a digraph is not smaller than its independence number, then the digraph is supereulerian. As a support for this conjecture we prove it for digraphs that are semicomplete multipartite or quasitransitive and verify the analogous statement for undirected graphs.     

Supereulerian bipartite digraphs

A digraph D is supereulerian if D has a spanning closed ditrail. Bang‐Jensen and Thomassé conjectured that if the arc‐strong connectivity of a digraph D is not less than the independence number , then D is supereulerian. A digraph is bipartite if its underlying graph is bipartite. Let be the size of a maximum matching of D. We prove that if D is a bipartite digraph satisfying , then D is supereulerian. Consequently, every bipartite digraph D satisfying is supereulerian. The bound of our main result is best possible.     

Supereulerian Digraphs with Large Arc‐Strong Connectivity

Let D be a digraph and let be the arc‐strong connectivity of D, and be the size of a maximum matching of D. We proved that if , then D has a spanning eulerian subdigraph.     

三边连通超欧拉图的一个注记

For two integers l 0 and k ≥ 0,define C(l,k) to be the family of 2-edge connected graphs such that a graph G ∈ C(l,k) if and only if for every bond S-E(G) with |S| ≤ 3,each component of G-S has order at least(|V(G)|-k)/l.In this note we prove that if a 3-edge-connected simple graph G is in C(10,3),then G is supereulerian if and only if G cannot be contracted to the Petersen graph.Our result extends an earlier result in [Supereulerian graphs and Petersen graph.JCMCC 1991,9:79-89] by Chen.     

Every 3‐connected claw‐free Z8‐free graph is Hamiltonian

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. Given a graph G, the biclique matrix of G is a {0,1,?1} matrix having one row for each biclique and one column for each vertex of G, and such that a pair of 1, ?1 entries in a same row corresponds exactly to adjacent vertices in the corresponding biclique. We describe a characterization of biclique matrices, in similar terms as those employed in Gilmore's characterization of clique matrices. On the other hand, the biclique graph of a graph is the intersection graph of the bicliques of G. Using the concept of biclique matrices, we describe a Krausz‐type characterization of biclique graphs. Finally, we show that every induced P₃ of a biclique graph must be included in a diamond or in a 3‐fan and we also characterize biclique graphs of bipartite graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 1–16, 2010     

超欧拉扩展有向图

A digraph D is supereulerian if D has a spanning eulerian subdigraph. BangJensen and Thomass′e conjectured that if the arc-strong connectivity λ(D) of a digraph D is not less than the independence number α(D), then D is supereulerian. In this paper, we prove that if D is an extended cycle, an extended hamiltonian digraph, an arc-locally semicomplete digraph, an extended arc-locally semicomplete digraph, an extension of two kinds of eulerian digraph, a hypo-semicomplete digraph or an extended hypo-semicomplete digraph satisfyingλ(D) ≥α(D), then D is supereulerian.     

有向跳图的生成欧拉子有向图

一个有向多重图D的跳图$J(D)$是一个顶点集为$D$的弧集,其中$(a,b)$是$J(D)$的一条弧当且仅当存在有向多重图$D$中的顶点$u_1$, $v_1$, $u_2$, $v_2$,使得$a=(u_1,v_1)$, $b=(u_2,v_2)$ 并且$v_1\neq u_2$.本文刻画了有向多重图类$\mathcal{H}_1$和$\mathcal{H}_2$,并证明了一个有向多重图$D$的跳图$J(D)$是强连通的当且仅当$D\not\in \mathcal{H}_1$.特别地, $J(D)$是弱连通的当且仅当$D\not\in \mathcal{H}_2$.进一步, 得到以下结果: (i) 存在有向多重图类$\mathcal{D}$使得有向多重图$D$的强连通跳图$J(D)$是强迹连通的当且仅当$D\not\in\mathcal{D}$. (ii) 每一个有向多重图$D$的强连通跳图$J(D)$是弱迹连通的,因此是超欧拉的. (iii) 每一个有向多重图D的弱连通跳图$J(D)$含有生成迹.     

关于超Euler图的一个猜想的注记

本文研究了Catlin的关于超Euler图的一个猜想,借助于收缩方法,得到了该猜想的两个充分条件.     

Supereulerian regular matroids without small cocircuits

A cycle of a matroid is a disjoint union of circuits. A matroid is supereulerian if it contains a spanning cycle. To answer an open problem of Bauer in 1985, Catlin proved in [J. Graph Theory 12 (1988) 29–44] that for sufficiently large $n$, every 2-edge-connected simple graph $G$ with $\text{◂=▸}n=\mid V\left(G\right)\mid$ and minimum degree $\text{◂≥▸}\delta \left(G\right)\ge \frac{n}{5}$ is supereulerian. In [Eur. J. Combinatorics, 33 (2012), 1765–1776], it is shown that for any connected simple regular matroid $M$, if every cocircuit $D$ of $M$ satisfies $\text{◂≥▸}\mid D\mid \ge \max \left\{\frac{\text{◂−▸}r\left(M\right)-5}{5},6\right\}$, then $M$ is supereulerian. We prove the following. (i) Let $M$ be a connected simple regular matroid. If every cocircuit $D$ of $M$ satisfies $\text{◂≥▸}\mid D\mid \ge \max \left\{\frac{\text{◂+▸}r\left(M\right)+1}{10},9\right\}$, then $M$ is supereulerian. (ii) For any real number $c$ with , there exists an integer $f\left(c\right)$ such that if every cocircuit $D$ of a connected simple cographic matroid $M$ satisfies $\mid D\mid \ge \text{◂lim▸}\max \text{◂{}▸}\{c\text{◂()▸}\left(r\left(M\right)+1\right),f\left(c\right)\}$, then $M$ is supereulerian.